Last updated: 2026-07-01

The harmonic mean

The harmonic mean of n positive numbers is the count divided by the sum of the reciprocals of the values. For values x₁, x₂, …, xₙ, the harmonic mean H is:

H = n / (1/x₁ + 1/x₂ + … + 1/xₙ)

Equivalently, it is the reciprocal of the arithmetic mean of the reciprocals. Where the arithmetic mean adds the values directly, the harmonic mean works in reciprocal space — which is exactly why it is the right average for rates.

When to use the harmonic mean

The harmonic mean is the appropriate average whenever the data are rates or ratios with a fixed quantity in the numerator. The most common applications are:

Average speed over equal distances. Speed is distance per unit time. When the distances covered at each speed are equal, the overall average speed is the harmonic mean of the individual speeds, not their arithmetic mean.

Price-to-earnings and other financial ratios. When averaging P/E ratios across companies — or building an index from them — the harmonic mean avoids the distortion that a few very high ratios would otherwise introduce in an arithmetic average.

Rates and densities in physics and engineering. Parallel resistances, effective rates of work for tasks done together, and fuel-economy figures (distance per unit of fuel) all combine through reciprocals, making the harmonic mean the correct summary.

A simple guideline: if the data are rates expressed as "something per unit" and the something is held constant, the harmonic mean gives the true overall rate.

Worked example

A delivery van drives a fixed route at 2 km/h, then 4 km/h, then 8 km/h over three equal-length segments. The average speed for the whole route is the harmonic mean of the three speeds:

• Sum of reciprocals: 1/2 + 1/4 + 1/8 = 0.875
• Harmonic mean: 3 / 0.875 = 24/7 ≈ 3.43 km/h

Compare this to the arithmetic mean of (2 + 4 + 8) / 3 ≈ 4.67 km/h. The arithmetic figure is too high because more time is spent on the slow segments — the harmonic mean correctly accounts for that extra time.

Why average speed is a harmonic mean

Consider a round trip of equal distances at 60 km/h out and 40 km/h back. The total distance is 2d and the total time is d/60 + d/40. Average speed is total distance over total time:

average speed = 2d / (d/60 + d/40) = 2 / (1/60 + 1/40) = 48 km/h

The distances d cancel, leaving exactly the harmonic mean of 60 and 40. Note that this works because the distances are equal. If instead the times were equal — driving 60 km/h for one hour, then 40 km/h for one hour — the arithmetic mean of 50 km/h would be correct. Choosing the right mean depends entirely on which quantity is held constant.

The AM–GM–HM inequality

For any list of positive numbers, the three Pythagorean means are ordered:

arithmetic mean ≥ geometric mean ≥ harmonic mean

Equality holds only when all values are identical. For the values 2, 4, 8 the arithmetic mean is about 4.667, the geometric mean is 4, and the harmonic mean is about 3.429 — a clear illustration of AM ≥ GM ≥ HM. The gaps widen as the data become more spread out, so the distance between the three means is itself a rough measure of dispersion.

This ordering follows from the convexity of the reciprocal and logarithm functions and is one of the classical inequalities of mathematics.

Why all values must be positive

The harmonic mean is built from reciprocals. A value of zero has no reciprocal, so the formula is undefined. A mix of positive and negative values can make the reciprocals nearly cancel, pushing the harmonic mean toward an arbitrarily large or meaningless number. In every real application — speeds, rates, prices, frequencies — the quantities are inherently positive, so requiring all values to be greater than zero matches practice.

Relationship to other means

The harmonic mean is one of three Pythagorean means:

• Arithmetic mean (AM): the ordinary average — appropriate for additive quantities.
• Geometric mean (GM): the nth root of the product — appropriate for multiplicative quantities such as growth factors.
• Harmonic mean (HM): the reciprocal of the average of reciprocals — appropriate for rates and ratios.

This calculator computes all three so you can compare them directly and confirm the AM ≥ GM ≥ HM ordering for your own data.

Related calculators

• Geometric Mean Calculator — nth root of the product, for multiplicative data
• Mean, Median, Mode Calculator — arithmetic mean, median, mode, and range
• Weighted Average Calculator — mean with per-value weights
• Descriptive Statistics Calculator — full summary statistics for a dataset